// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_PRODUCTEVALUATORS_H
#define EIGEN_PRODUCTEVALUATORS_H

namespace Eigen {

namespace internal {

    /** \internal
  * Evaluator of a product expression.
  * Since products require special treatments to handle all possible cases,
  * we simply defer the evaluation logic to a product_evaluator class
  * which offers more partial specialization possibilities.
  *
  * \sa class product_evaluator
  */
    template <typename Lhs, typename Rhs, int Options> struct evaluator<Product<Lhs, Rhs, Options>> : public product_evaluator<Product<Lhs, Rhs, Options>>
    {
        typedef Product<Lhs, Rhs, Options> XprType;
        typedef product_evaluator<XprType> Base;

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit evaluator(const XprType& xpr) : Base(xpr) {}
    };

    // Catch "scalar * ( A * B )" and transform it to "(A*scalar) * B"
    // TODO we should apply that rule only if that's really helpful
    template <typename Lhs, typename Rhs, typename Scalar1, typename Scalar2, typename Plain1>
    struct evaluator_assume_aliasing<CwiseBinaryOp<internal::scalar_product_op<Scalar1, Scalar2>,
                                                   const CwiseNullaryOp<internal::scalar_constant_op<Scalar1>, Plain1>,
                                                   const Product<Lhs, Rhs, DefaultProduct>>>
    {
        static const bool value = true;
    };
    template <typename Lhs, typename Rhs, typename Scalar1, typename Scalar2, typename Plain1>
    struct evaluator<CwiseBinaryOp<internal::scalar_product_op<Scalar1, Scalar2>,
                                   const CwiseNullaryOp<internal::scalar_constant_op<Scalar1>, Plain1>,
                                   const Product<Lhs, Rhs, DefaultProduct>>>
        : public evaluator<Product<EIGEN_SCALAR_BINARYOP_EXPR_RETURN_TYPE(Scalar1, Lhs, product), Rhs, DefaultProduct>>
    {
        typedef CwiseBinaryOp<internal::scalar_product_op<Scalar1, Scalar2>,
                              const CwiseNullaryOp<internal::scalar_constant_op<Scalar1>, Plain1>,
                              const Product<Lhs, Rhs, DefaultProduct>>
            XprType;
        typedef evaluator<Product<EIGEN_SCALAR_BINARYOP_EXPR_RETURN_TYPE(Scalar1, Lhs, product), Rhs, DefaultProduct>> Base;

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit evaluator(const XprType& xpr) : Base(xpr.lhs().functor().m_other * xpr.rhs().lhs() * xpr.rhs().rhs()) {}
    };

    template <typename Lhs, typename Rhs, int DiagIndex>
    struct evaluator<Diagonal<const Product<Lhs, Rhs, DefaultProduct>, DiagIndex>> : public evaluator<Diagonal<const Product<Lhs, Rhs, LazyProduct>, DiagIndex>>
    {
        typedef Diagonal<const Product<Lhs, Rhs, DefaultProduct>, DiagIndex> XprType;
        typedef evaluator<Diagonal<const Product<Lhs, Rhs, LazyProduct>, DiagIndex>> Base;

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit evaluator(const XprType& xpr)
            : Base(Diagonal<const Product<Lhs, Rhs, LazyProduct>, DiagIndex>(
                  Product<Lhs, Rhs, LazyProduct>(xpr.nestedExpression().lhs(), xpr.nestedExpression().rhs()),
                  xpr.index()))
        {
        }
    };

    // Helper class to perform a matrix product with the destination at hand.
    // Depending on the sizes of the factors, there are different evaluation strategies
    // as controlled by internal::product_type.
    template <typename Lhs,
              typename Rhs,
              typename LhsShape = typename evaluator_traits<Lhs>::Shape,
              typename RhsShape = typename evaluator_traits<Rhs>::Shape,
              int ProductType = internal::product_type<Lhs, Rhs>::value>
    struct generic_product_impl;

    template <typename Lhs, typename Rhs> struct evaluator_assume_aliasing<Product<Lhs, Rhs, DefaultProduct>>
    {
        static const bool value = true;
    };

    // This is the default evaluator implementation for products:
    // It creates a temporary and call generic_product_impl
    template <typename Lhs, typename Rhs, int Options, int ProductTag, typename LhsShape, typename RhsShape>
    struct product_evaluator<Product<Lhs, Rhs, Options>, ProductTag, LhsShape, RhsShape> : public evaluator<typename Product<Lhs, Rhs, Options>::PlainObject>
    {
        typedef Product<Lhs, Rhs, Options> XprType;
        typedef typename XprType::PlainObject PlainObject;
        typedef evaluator<PlainObject> Base;
        enum
        {
            Flags = Base::Flags | EvalBeforeNestingBit
        };

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit product_evaluator(const XprType& xpr) : m_result(xpr.rows(), xpr.cols())
        {
            ::new (static_cast<Base*>(this)) Base(m_result);

            // FIXME shall we handle nested_eval here?,
            // if so, then we must take care at removing the call to nested_eval in the specializations (e.g., in permutation_matrix_product, transposition_matrix_product, etc.)
            //     typedef typename internal::nested_eval<Lhs,Rhs::ColsAtCompileTime>::type LhsNested;
            //     typedef typename internal::nested_eval<Rhs,Lhs::RowsAtCompileTime>::type RhsNested;
            //     typedef typename internal::remove_all<LhsNested>::type LhsNestedCleaned;
            //     typedef typename internal::remove_all<RhsNested>::type RhsNestedCleaned;
            //
            //     const LhsNested lhs(xpr.lhs());
            //     const RhsNested rhs(xpr.rhs());
            //
            //     generic_product_impl<LhsNestedCleaned, RhsNestedCleaned>::evalTo(m_result, lhs, rhs);

            generic_product_impl<Lhs, Rhs, LhsShape, RhsShape, ProductTag>::evalTo(m_result, xpr.lhs(), xpr.rhs());
        }

    protected:
        PlainObject m_result;
    };

    // The following three shortcuts are enabled only if the scalar types match exactly.
    // TODO: we could enable them for different scalar types when the product is not vectorized.

    // Dense = Product
    template <typename DstXprType, typename Lhs, typename Rhs, int Options, typename Scalar>
    struct Assignment<DstXprType,
                      Product<Lhs, Rhs, Options>,
                      internal::assign_op<Scalar, Scalar>,
                      Dense2Dense,
                      typename enable_if<(Options == DefaultProduct || Options == AliasFreeProduct)>::type>
    {
        typedef Product<Lhs, Rhs, Options> SrcXprType;
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src, const internal::assign_op<Scalar, Scalar>&)
        {
            Index dstRows = src.rows();
            Index dstCols = src.cols();
            if ((dst.rows() != dstRows) || (dst.cols() != dstCols))
                dst.resize(dstRows, dstCols);
            // FIXME shall we handle nested_eval here?
            generic_product_impl<Lhs, Rhs>::evalTo(dst, src.lhs(), src.rhs());
        }
    };

    // Dense += Product
    template <typename DstXprType, typename Lhs, typename Rhs, int Options, typename Scalar>
    struct Assignment<DstXprType,
                      Product<Lhs, Rhs, Options>,
                      internal::add_assign_op<Scalar, Scalar>,
                      Dense2Dense,
                      typename enable_if<(Options == DefaultProduct || Options == AliasFreeProduct)>::type>
    {
        typedef Product<Lhs, Rhs, Options> SrcXprType;
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src, const internal::add_assign_op<Scalar, Scalar>&)
        {
            eigen_assert(dst.rows() == src.rows() && dst.cols() == src.cols());
            // FIXME shall we handle nested_eval here?
            generic_product_impl<Lhs, Rhs>::addTo(dst, src.lhs(), src.rhs());
        }
    };

    // Dense -= Product
    template <typename DstXprType, typename Lhs, typename Rhs, int Options, typename Scalar>
    struct Assignment<DstXprType,
                      Product<Lhs, Rhs, Options>,
                      internal::sub_assign_op<Scalar, Scalar>,
                      Dense2Dense,
                      typename enable_if<(Options == DefaultProduct || Options == AliasFreeProduct)>::type>
    {
        typedef Product<Lhs, Rhs, Options> SrcXprType;
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src, const internal::sub_assign_op<Scalar, Scalar>&)
        {
            eigen_assert(dst.rows() == src.rows() && dst.cols() == src.cols());
            // FIXME shall we handle nested_eval here?
            generic_product_impl<Lhs, Rhs>::subTo(dst, src.lhs(), src.rhs());
        }
    };

    // Dense ?= scalar * Product
    // TODO we should apply that rule if that's really helpful
    // for instance, this is not good for inner products
    template <typename DstXprType, typename Lhs, typename Rhs, typename AssignFunc, typename Scalar, typename ScalarBis, typename Plain>
    struct Assignment<DstXprType,
                      CwiseBinaryOp<internal::scalar_product_op<ScalarBis, Scalar>,
                                    const CwiseNullaryOp<internal::scalar_constant_op<ScalarBis>, Plain>,
                                    const Product<Lhs, Rhs, DefaultProduct>>,
                      AssignFunc,
                      Dense2Dense>
    {
        typedef CwiseBinaryOp<internal::scalar_product_op<ScalarBis, Scalar>,
                              const CwiseNullaryOp<internal::scalar_constant_op<ScalarBis>, Plain>,
                              const Product<Lhs, Rhs, DefaultProduct>>
            SrcXprType;
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src, const AssignFunc& func)
        {
            call_assignment_no_alias(dst, (src.lhs().functor().m_other * src.rhs().lhs()) * src.rhs().rhs(), func);
        }
    };

    //----------------------------------------
    // Catch "Dense ?= xpr + Product<>" expression to save one temporary
    // FIXME we could probably enable these rules for any product, i.e., not only Dense and DefaultProduct

    template <typename OtherXpr, typename Lhs, typename Rhs>
    struct evaluator_assume_aliasing<CwiseBinaryOp<internal::scalar_sum_op<typename OtherXpr::Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
                                                   const OtherXpr,
                                                   const Product<Lhs, Rhs, DefaultProduct>>,
                                     DenseShape>
    {
        static const bool value = true;
    };

    template <typename OtherXpr, typename Lhs, typename Rhs>
    struct evaluator_assume_aliasing<
        CwiseBinaryOp<internal::scalar_difference_op<typename OtherXpr::Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
                      const OtherXpr,
                      const Product<Lhs, Rhs, DefaultProduct>>,
        DenseShape>
    {
        static const bool value = true;
    };

    template <typename DstXprType, typename OtherXpr, typename ProductType, typename Func1, typename Func2> struct assignment_from_xpr_op_product
    {
        template <typename SrcXprType, typename InitialFunc>
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(DstXprType& dst, const SrcXprType& src, const InitialFunc& /*func*/)
        {
            call_assignment_no_alias(dst, src.lhs(), Func1());
            call_assignment_no_alias(dst, src.rhs(), Func2());
        }
    };

#define EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(ASSIGN_OP, BINOP, ASSIGN_OP2)                                                                 \
    template <typename DstXprType,                                                                                                      \
              typename OtherXpr,                                                                                                        \
              typename Lhs,                                                                                                             \
              typename Rhs,                                                                                                             \
              typename DstScalar,                                                                                                       \
              typename SrcScalar,                                                                                                       \
              typename OtherScalar,                                                                                                     \
              typename ProdScalar>                                                                                                      \
    struct Assignment<DstXprType,                                                                                                       \
                      CwiseBinaryOp<internal::BINOP<OtherScalar, ProdScalar>, const OtherXpr, const Product<Lhs, Rhs, DefaultProduct>>, \
                      internal::ASSIGN_OP<DstScalar, SrcScalar>,                                                                        \
                      Dense2Dense> : assignment_from_xpr_op_product<DstXprType,                                                         \
                                                                    OtherXpr,                                                           \
                                                                    Product<Lhs, Rhs, DefaultProduct>,                                  \
                                                                    internal::ASSIGN_OP<DstScalar, OtherScalar>,                        \
                                                                    internal::ASSIGN_OP2<DstScalar, ProdScalar>>                        \
    {                                                                                                                                   \
    }

    EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(assign_op, scalar_sum_op, add_assign_op);
    EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(add_assign_op, scalar_sum_op, add_assign_op);
    EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(sub_assign_op, scalar_sum_op, sub_assign_op);

    EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(assign_op, scalar_difference_op, sub_assign_op);
    EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(add_assign_op, scalar_difference_op, sub_assign_op);
    EIGEN_CATCH_ASSIGN_XPR_OP_PRODUCT(sub_assign_op, scalar_difference_op, add_assign_op);

    //----------------------------------------

    template <typename Lhs, typename Rhs> struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, InnerProduct>
    {
        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            dst.coeffRef(0, 0) = (lhs.transpose().cwiseProduct(rhs)).sum();
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            dst.coeffRef(0, 0) += (lhs.transpose().cwiseProduct(rhs)).sum();
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            dst.coeffRef(0, 0) -= (lhs.transpose().cwiseProduct(rhs)).sum();
        }
    };

    /***********************************************************************
*  Implementation of outer dense * dense vector product
***********************************************************************/

    // Column major result
    template <typename Dst, typename Lhs, typename Rhs, typename Func>
    void EIGEN_DEVICE_FUNC outer_product_selector_run(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Func& func, const false_type&)
    {
        evaluator<Rhs> rhsEval(rhs);
        ei_declare_local_nested_eval(Lhs, lhs, Rhs::SizeAtCompileTime, actual_lhs);
        // FIXME if cols is large enough, then it might be useful to make sure that lhs is sequentially stored
        // FIXME not very good if rhs is real and lhs complex while alpha is real too
        const Index cols = dst.cols();
        for (Index j = 0; j < cols; ++j) func(dst.col(j), rhsEval.coeff(Index(0), j) * actual_lhs);
    }

    // Row major result
    template <typename Dst, typename Lhs, typename Rhs, typename Func>
    void EIGEN_DEVICE_FUNC outer_product_selector_run(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Func& func, const true_type&)
    {
        evaluator<Lhs> lhsEval(lhs);
        ei_declare_local_nested_eval(Rhs, rhs, Lhs::SizeAtCompileTime, actual_rhs);
        // FIXME if rows is large enough, then it might be useful to make sure that rhs is sequentially stored
        // FIXME not very good if lhs is real and rhs complex while alpha is real too
        const Index rows = dst.rows();
        for (Index i = 0; i < rows; ++i) func(dst.row(i), lhsEval.coeff(i, Index(0)) * actual_rhs);
    }

    template <typename Lhs, typename Rhs> struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, OuterProduct>
    {
        template <typename T> struct is_row_major : internal::conditional<(int(T::Flags) & RowMajorBit), internal::true_type, internal::false_type>::type
        {
        };
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        // TODO it would be nice to be able to exploit our *_assign_op functors for that purpose
        struct set
        {
            template <typename Dst, typename Src> EIGEN_DEVICE_FUNC void operator()(const Dst& dst, const Src& src) const { dst.const_cast_derived() = src; }
        };
        struct add
        {
            template <typename Dst, typename Src> EIGEN_DEVICE_FUNC void operator()(const Dst& dst, const Src& src) const { dst.const_cast_derived() += src; }
        };
        struct sub
        {
            template <typename Dst, typename Src> EIGEN_DEVICE_FUNC void operator()(const Dst& dst, const Src& src) const { dst.const_cast_derived() -= src; }
        };
        struct adds
        {
            Scalar m_scale;
            explicit adds(const Scalar& s) : m_scale(s) {}
            template <typename Dst, typename Src> void EIGEN_DEVICE_FUNC operator()(const Dst& dst, const Src& src) const
            {
                dst.const_cast_derived() += m_scale * src;
            }
        };

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            internal::outer_product_selector_run(dst, lhs, rhs, set(), is_row_major<Dst>());
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            internal::outer_product_selector_run(dst, lhs, rhs, add(), is_row_major<Dst>());
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            internal::outer_product_selector_run(dst, lhs, rhs, sub(), is_row_major<Dst>());
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void scaleAndAddTo(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            internal::outer_product_selector_run(dst, lhs, rhs, adds(alpha), is_row_major<Dst>());
        }
    };

    // This base class provides default implementations for evalTo, addTo, subTo, in terms of scaleAndAddTo
    template <typename Lhs, typename Rhs, typename Derived> struct generic_product_impl_base
    {
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            dst.setZero();
            scaleAndAddTo(dst, lhs, rhs, Scalar(1));
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            scaleAndAddTo(dst, lhs, rhs, Scalar(1));
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            scaleAndAddTo(dst, lhs, rhs, Scalar(-1));
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void scaleAndAddTo(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            Derived::scaleAndAddTo(dst, lhs, rhs, alpha);
        }
    };

    template <typename Lhs, typename Rhs>
    struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, GemvProduct>
        : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, GemvProduct>>
    {
        typedef typename nested_eval<Lhs, 1>::type LhsNested;
        typedef typename nested_eval<Rhs, 1>::type RhsNested;
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;
        enum
        {
            Side = Lhs::IsVectorAtCompileTime ? OnTheLeft : OnTheRight
        };
        typedef typename internal::remove_all<typename internal::conditional<int(Side) == OnTheRight, LhsNested, RhsNested>::type>::type MatrixType;

        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            // Fallback to inner product if both the lhs and rhs is a runtime vector.
            if (lhs.rows() == 1 && rhs.cols() == 1)
            {
                dst.coeffRef(0, 0) += alpha * lhs.row(0).conjugate().dot(rhs.col(0));
                return;
            }
            LhsNested actual_lhs(lhs);
            RhsNested actual_rhs(rhs);
            internal::gemv_dense_selector<Side,
                                          (int(MatrixType::Flags) & RowMajorBit) ? RowMajor : ColMajor,
                                          bool(internal::blas_traits<MatrixType>::HasUsableDirectAccess)>::run(actual_lhs, actual_rhs, dst, alpha);
        }
    };

    template <typename Lhs, typename Rhs> struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, CoeffBasedProductMode>
    {
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            // Same as: dst.noalias() = lhs.lazyProduct(rhs);
            // but easier on the compiler side
            call_assignment_no_alias(dst, lhs.lazyProduct(rhs), internal::assign_op<typename Dst::Scalar, Scalar>());
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void addTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            // dst.noalias() += lhs.lazyProduct(rhs);
            call_assignment_no_alias(dst, lhs.lazyProduct(rhs), internal::add_assign_op<typename Dst::Scalar, Scalar>());
        }

        template <typename Dst> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void subTo(Dst& dst, const Lhs& lhs, const Rhs& rhs)
        {
            // dst.noalias() -= lhs.lazyProduct(rhs);
            call_assignment_no_alias(dst, lhs.lazyProduct(rhs), internal::sub_assign_op<typename Dst::Scalar, Scalar>());
        }

        // This is a special evaluation path called from generic_product_impl<...,GemmProduct> in file GeneralMatrixMatrix.h
        // This variant tries to extract scalar multiples from both the LHS and RHS and factor them out. For instance:
        //   dst {,+,-}= (s1*A)*(B*s2)
        // will be rewritten as:
        //   dst {,+,-}= (s1*s2) * (A.lazyProduct(B))
        // There are at least four benefits of doing so:
        //  1 - huge performance gain for heap-allocated matrix types as it save costly allocations.
        //  2 - it is faster than simply by-passing the heap allocation through stack allocation.
        //  3 - it makes this fallback consistent with the heavy GEMM routine.
        //  4 - it fully by-passes huge stack allocation attempts when multiplying huge fixed-size matrices.
        //      (see https://stackoverflow.com/questions/54738495)
        // For small fixed sizes matrices, howver, the gains are less obvious, it is sometimes x2 faster, but sometimes x3 slower,
        // and the behavior depends also a lot on the compiler... This is why this re-writting strategy is currently
        // enabled only when falling back from the main GEMM.
        template <typename Dst, typename Func>
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void eval_dynamic(Dst& dst, const Lhs& lhs, const Rhs& rhs, const Func& func)
        {
            enum
            {
                HasScalarFactor = blas_traits<Lhs>::HasScalarFactor || blas_traits<Rhs>::HasScalarFactor,
                ConjLhs = blas_traits<Lhs>::NeedToConjugate,
                ConjRhs = blas_traits<Rhs>::NeedToConjugate
            };
            // FIXME: in c++11 this should be auto, and extractScalarFactor should also return auto
            //        this is important for real*complex_mat
            Scalar actualAlpha = combine_scalar_factors<Scalar>(lhs, rhs);

            eval_dynamic_impl(dst,
                              blas_traits<Lhs>::extract(lhs).template conjugateIf<ConjLhs>(),
                              blas_traits<Rhs>::extract(rhs).template conjugateIf<ConjRhs>(),
                              func,
                              actualAlpha,
                              typename conditional<HasScalarFactor, true_type, false_type>::type());
        }

    protected:
        template <typename Dst, typename LhsT, typename RhsT, typename Func, typename Scalar>
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void
        eval_dynamic_impl(Dst& dst, const LhsT& lhs, const RhsT& rhs, const Func& func, const Scalar& s /* == 1 */, false_type)
        {
            EIGEN_UNUSED_VARIABLE(s);
            eigen_internal_assert(s == Scalar(1));
            call_restricted_packet_assignment_no_alias(dst, lhs.lazyProduct(rhs), func);
        }

        template <typename Dst, typename LhsT, typename RhsT, typename Func, typename Scalar>
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void
        eval_dynamic_impl(Dst& dst, const LhsT& lhs, const RhsT& rhs, const Func& func, const Scalar& s, true_type)
        {
            call_restricted_packet_assignment_no_alias(dst, s * lhs.lazyProduct(rhs), func);
        }
    };

    // This specialization enforces the use of a coefficient-based evaluation strategy
    template <typename Lhs, typename Rhs>
    struct generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, LazyCoeffBasedProductMode>
        : generic_product_impl<Lhs, Rhs, DenseShape, DenseShape, CoeffBasedProductMode>
    {
    };

    // Case 2: Evaluate coeff by coeff
    //
    // This is mostly taken from CoeffBasedProduct.h
    // The main difference is that we add an extra argument to the etor_product_*_impl::run() function
    // for the inner dimension of the product, because evaluator object do not know their size.

    template <int Traversal, int UnrollingIndex, typename Lhs, typename Rhs, typename RetScalar> struct etor_product_coeff_impl;

    template <int StorageOrder, int UnrollingIndex, typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl;

    template <typename Lhs, typename Rhs, int ProductTag>
    struct product_evaluator<Product<Lhs, Rhs, LazyProduct>, ProductTag, DenseShape, DenseShape> : evaluator_base<Product<Lhs, Rhs, LazyProduct>>
    {
        typedef Product<Lhs, Rhs, LazyProduct> XprType;
        typedef typename XprType::Scalar Scalar;
        typedef typename XprType::CoeffReturnType CoeffReturnType;

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit product_evaluator(const XprType& xpr)
            : m_lhs(xpr.lhs()), m_rhs(xpr.rhs()), m_lhsImpl(m_lhs),  // FIXME the creation of the evaluator objects should result in a no-op, but check that!
              m_rhsImpl(m_rhs),  //       Moreover, they are only useful for the packet path, so we could completely disable them when not needed,
                                 //       or perhaps declare them on the fly on the packet method... We have experiment to check what's best.
              m_innerDim(xpr.lhs().cols())
        {
            EIGEN_INTERNAL_CHECK_COST_VALUE(NumTraits<Scalar>::MulCost);
            EIGEN_INTERNAL_CHECK_COST_VALUE(NumTraits<Scalar>::AddCost);
            EIGEN_INTERNAL_CHECK_COST_VALUE(CoeffReadCost);
#if 0
    std::cerr << "LhsOuterStrideBytes=  " << LhsOuterStrideBytes << "\n";
    std::cerr << "RhsOuterStrideBytes=  " << RhsOuterStrideBytes << "\n";
    std::cerr << "LhsAlignment=         " << LhsAlignment << "\n";
    std::cerr << "RhsAlignment=         " << RhsAlignment << "\n";
    std::cerr << "CanVectorizeLhs=      " << CanVectorizeLhs << "\n";
    std::cerr << "CanVectorizeRhs=      " << CanVectorizeRhs << "\n";
    std::cerr << "CanVectorizeInner=    " << CanVectorizeInner << "\n";
    std::cerr << "EvalToRowMajor=       " << EvalToRowMajor << "\n";
    std::cerr << "Alignment=            " << Alignment << "\n";
    std::cerr << "Flags=                " << Flags << "\n";
#endif
        }

        // Everything below here is taken from CoeffBasedProduct.h

        typedef typename internal::nested_eval<Lhs, Rhs::ColsAtCompileTime>::type LhsNested;
        typedef typename internal::nested_eval<Rhs, Lhs::RowsAtCompileTime>::type RhsNested;

        typedef typename internal::remove_all<LhsNested>::type LhsNestedCleaned;
        typedef typename internal::remove_all<RhsNested>::type RhsNestedCleaned;

        typedef evaluator<LhsNestedCleaned> LhsEtorType;
        typedef evaluator<RhsNestedCleaned> RhsEtorType;

        enum
        {
            RowsAtCompileTime = LhsNestedCleaned::RowsAtCompileTime,
            ColsAtCompileTime = RhsNestedCleaned::ColsAtCompileTime,
            InnerSize = EIGEN_SIZE_MIN_PREFER_FIXED(LhsNestedCleaned::ColsAtCompileTime, RhsNestedCleaned::RowsAtCompileTime),
            MaxRowsAtCompileTime = LhsNestedCleaned::MaxRowsAtCompileTime,
            MaxColsAtCompileTime = RhsNestedCleaned::MaxColsAtCompileTime
        };

        typedef typename find_best_packet<Scalar, RowsAtCompileTime>::type LhsVecPacketType;
        typedef typename find_best_packet<Scalar, ColsAtCompileTime>::type RhsVecPacketType;

        enum
        {

            LhsCoeffReadCost = LhsEtorType::CoeffReadCost,
            RhsCoeffReadCost = RhsEtorType::CoeffReadCost,
            CoeffReadCost = InnerSize == 0 ? NumTraits<Scalar>::ReadCost :
                                             InnerSize == Dynamic ? HugeCost :
                                                                    InnerSize * (NumTraits<Scalar>::MulCost + int(LhsCoeffReadCost) + int(RhsCoeffReadCost)) +
                                                                        (InnerSize - 1) * NumTraits<Scalar>::AddCost,

            Unroll = CoeffReadCost <= EIGEN_UNROLLING_LIMIT,

            LhsFlags = LhsEtorType::Flags,
            RhsFlags = RhsEtorType::Flags,

            LhsRowMajor = LhsFlags & RowMajorBit,
            RhsRowMajor = RhsFlags & RowMajorBit,

            LhsVecPacketSize = unpacket_traits<LhsVecPacketType>::size,
            RhsVecPacketSize = unpacket_traits<RhsVecPacketType>::size,

            // Here, we don't care about alignment larger than the usable packet size.
            LhsAlignment = EIGEN_PLAIN_ENUM_MIN(LhsEtorType::Alignment, LhsVecPacketSize* int(sizeof(typename LhsNestedCleaned::Scalar))),
            RhsAlignment = EIGEN_PLAIN_ENUM_MIN(RhsEtorType::Alignment, RhsVecPacketSize* int(sizeof(typename RhsNestedCleaned::Scalar))),

            SameType = is_same<typename LhsNestedCleaned::Scalar, typename RhsNestedCleaned::Scalar>::value,

            CanVectorizeRhs = bool(RhsRowMajor) && (RhsFlags & PacketAccessBit) && (ColsAtCompileTime != 1),
            CanVectorizeLhs = (!LhsRowMajor) && (LhsFlags & PacketAccessBit) && (RowsAtCompileTime != 1),

            EvalToRowMajor = (MaxRowsAtCompileTime == 1 && MaxColsAtCompileTime != 1) ?
                                 1 :
                                 (MaxColsAtCompileTime == 1 && MaxRowsAtCompileTime != 1) ? 0 : (bool(RhsRowMajor) && !CanVectorizeLhs),

            Flags = ((int(LhsFlags) | int(RhsFlags)) & HereditaryBits & ~RowMajorBit) |
                    (EvalToRowMajor ? RowMajorBit : 0)
                    // TODO enable vectorization for mixed types
                    | (SameType && (CanVectorizeLhs || CanVectorizeRhs) ? PacketAccessBit : 0) | (XprType::IsVectorAtCompileTime ? LinearAccessBit : 0),

            LhsOuterStrideBytes = int(LhsNestedCleaned::OuterStrideAtCompileTime) * int(sizeof(typename LhsNestedCleaned::Scalar)),
            RhsOuterStrideBytes = int(RhsNestedCleaned::OuterStrideAtCompileTime) * int(sizeof(typename RhsNestedCleaned::Scalar)),

            Alignment = bool(CanVectorizeLhs) ?
                            (LhsOuterStrideBytes <= 0 || (int(LhsOuterStrideBytes) % EIGEN_PLAIN_ENUM_MAX(1, LhsAlignment)) != 0 ? 0 : LhsAlignment) :
                            bool(CanVectorizeRhs) ?
                            (RhsOuterStrideBytes <= 0 || (int(RhsOuterStrideBytes) % EIGEN_PLAIN_ENUM_MAX(1, RhsAlignment)) != 0 ? 0 : RhsAlignment) :
                            0,

            /* CanVectorizeInner deserves special explanation. It does not affect the product flags. It is not used outside
     * of Product. If the Product itself is not a packet-access expression, there is still a chance that the inner
     * loop of the product might be vectorized. This is the meaning of CanVectorizeInner. Since it doesn't affect
     * the Flags, it is safe to make this value depend on ActualPacketAccessBit, that doesn't affect the ABI.
     */
            CanVectorizeInner = SameType && LhsRowMajor && (!RhsRowMajor) && (int(LhsFlags) & int(RhsFlags) & ActualPacketAccessBit) &&
                                (int(InnerSize) % packet_traits<Scalar>::size == 0)
        };

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const CoeffReturnType coeff(Index row, Index col) const
        {
            return (m_lhs.row(row).transpose().cwiseProduct(m_rhs.col(col))).sum();
        }

        /* Allow index-based non-packet access. It is impossible though to allow index-based packed access,
   * which is why we don't set the LinearAccessBit.
   * TODO: this seems possible when the result is a vector
   */
        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const CoeffReturnType coeff(Index index) const
        {
            const Index row = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? 0 : index;
            const Index col = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? index : 0;
            return (m_lhs.row(row).transpose().cwiseProduct(m_rhs.col(col))).sum();
        }

        template <int LoadMode, typename PacketType> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const PacketType packet(Index row, Index col) const
        {
            PacketType res;
            typedef etor_product_packet_impl<bool(int(Flags) & RowMajorBit) ? RowMajor : ColMajor,
                                             Unroll ? int(InnerSize) : Dynamic,
                                             LhsEtorType,
                                             RhsEtorType,
                                             PacketType,
                                             LoadMode>
                PacketImpl;
            PacketImpl::run(row, col, m_lhsImpl, m_rhsImpl, m_innerDim, res);
            return res;
        }

        template <int LoadMode, typename PacketType> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const PacketType packet(Index index) const
        {
            const Index row = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? 0 : index;
            const Index col = (RowsAtCompileTime == 1 || MaxRowsAtCompileTime == 1) ? index : 0;
            return packet<LoadMode, PacketType>(row, col);
        }

    protected:
        typename internal::add_const_on_value_type<LhsNested>::type m_lhs;
        typename internal::add_const_on_value_type<RhsNested>::type m_rhs;

        LhsEtorType m_lhsImpl;
        RhsEtorType m_rhsImpl;

        // TODO: Get rid of m_innerDim if known at compile time
        Index m_innerDim;
    };

    template <typename Lhs, typename Rhs>
    struct product_evaluator<Product<Lhs, Rhs, DefaultProduct>, LazyCoeffBasedProductMode, DenseShape, DenseShape>
        : product_evaluator<Product<Lhs, Rhs, LazyProduct>, CoeffBasedProductMode, DenseShape, DenseShape>
    {
        typedef Product<Lhs, Rhs, DefaultProduct> XprType;
        typedef Product<Lhs, Rhs, LazyProduct> BaseProduct;
        typedef product_evaluator<BaseProduct, CoeffBasedProductMode, DenseShape, DenseShape> Base;
        enum
        {
            Flags = Base::Flags | EvalBeforeNestingBit
        };
        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE explicit product_evaluator(const XprType& xpr) : Base(BaseProduct(xpr.lhs(), xpr.rhs())) {}
    };

    /****************************************
*** Coeff based product, Packet path  ***
****************************************/

    template <int UnrollingIndex, typename Lhs, typename Rhs, typename Packet, int LoadMode>
    struct etor_product_packet_impl<RowMajor, UnrollingIndex, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs, Index innerDim, Packet& res)
        {
            etor_product_packet_impl<RowMajor, UnrollingIndex - 1, Lhs, Rhs, Packet, LoadMode>::run(row, col, lhs, rhs, innerDim, res);
            res = pmadd(pset1<Packet>(lhs.coeff(row, Index(UnrollingIndex - 1))), rhs.template packet<LoadMode, Packet>(Index(UnrollingIndex - 1), col), res);
        }
    };

    template <int UnrollingIndex, typename Lhs, typename Rhs, typename Packet, int LoadMode>
    struct etor_product_packet_impl<ColMajor, UnrollingIndex, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs, Index innerDim, Packet& res)
        {
            etor_product_packet_impl<ColMajor, UnrollingIndex - 1, Lhs, Rhs, Packet, LoadMode>::run(row, col, lhs, rhs, innerDim, res);
            res = pmadd(lhs.template packet<LoadMode, Packet>(row, Index(UnrollingIndex - 1)), pset1<Packet>(rhs.coeff(Index(UnrollingIndex - 1), col)), res);
        }
    };

    template <typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl<RowMajor, 1, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs, Index /*innerDim*/, Packet& res)
        {
            res = pmul(pset1<Packet>(lhs.coeff(row, Index(0))), rhs.template packet<LoadMode, Packet>(Index(0), col));
        }
    };

    template <typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl<ColMajor, 1, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs, Index /*innerDim*/, Packet& res)
        {
            res = pmul(lhs.template packet<LoadMode, Packet>(row, Index(0)), pset1<Packet>(rhs.coeff(Index(0), col)));
        }
    };

    template <typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl<RowMajor, 0, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void
        run(Index /*row*/, Index /*col*/, const Lhs& /*lhs*/, const Rhs& /*rhs*/, Index /*innerDim*/, Packet& res)
        {
            res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
        }
    };

    template <typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl<ColMajor, 0, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void
        run(Index /*row*/, Index /*col*/, const Lhs& /*lhs*/, const Rhs& /*rhs*/, Index /*innerDim*/, Packet& res)
        {
            res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
        }
    };

    template <typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl<RowMajor, Dynamic, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs, Index innerDim, Packet& res)
        {
            res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
            for (Index i = 0; i < innerDim; ++i) res = pmadd(pset1<Packet>(lhs.coeff(row, i)), rhs.template packet<LoadMode, Packet>(i, col), res);
        }
    };

    template <typename Lhs, typename Rhs, typename Packet, int LoadMode> struct etor_product_packet_impl<ColMajor, Dynamic, Lhs, Rhs, Packet, LoadMode>
    {
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Index row, Index col, const Lhs& lhs, const Rhs& rhs, Index innerDim, Packet& res)
        {
            res = pset1<Packet>(typename unpacket_traits<Packet>::type(0));
            for (Index i = 0; i < innerDim; ++i) res = pmadd(lhs.template packet<LoadMode, Packet>(row, i), pset1<Packet>(rhs.coeff(i, col)), res);
        }
    };

    /***************************************************************************
* Triangular products
***************************************************************************/
    template <int Mode, bool LhsIsTriangular, typename Lhs, bool LhsIsVector, typename Rhs, bool RhsIsVector> struct triangular_product_impl;

    template <typename Lhs, typename Rhs, int ProductTag>
    struct generic_product_impl<Lhs, Rhs, TriangularShape, DenseShape, ProductTag>
        : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, TriangularShape, DenseShape, ProductTag>>
    {
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        template <typename Dest> static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            triangular_product_impl<Lhs::Mode, true, typename Lhs::MatrixType, false, Rhs, Rhs::ColsAtCompileTime == 1>::run(
                dst, lhs.nestedExpression(), rhs, alpha);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag>
    struct generic_product_impl<Lhs, Rhs, DenseShape, TriangularShape, ProductTag>
        : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, TriangularShape, ProductTag>>
    {
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        template <typename Dest> static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            triangular_product_impl<Rhs::Mode, false, Lhs, Lhs::RowsAtCompileTime == 1, typename Rhs::MatrixType, false>::run(
                dst, lhs, rhs.nestedExpression(), alpha);
        }
    };

    /***************************************************************************
* SelfAdjoint products
***************************************************************************/
    template <typename Lhs, int LhsMode, bool LhsIsVector, typename Rhs, int RhsMode, bool RhsIsVector> struct selfadjoint_product_impl;

    template <typename Lhs, typename Rhs, int ProductTag>
    struct generic_product_impl<Lhs, Rhs, SelfAdjointShape, DenseShape, ProductTag>
        : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, SelfAdjointShape, DenseShape, ProductTag>>
    {
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        template <typename Dest> static EIGEN_DEVICE_FUNC void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            selfadjoint_product_impl<typename Lhs::MatrixType, Lhs::Mode, false, Rhs, 0, Rhs::IsVectorAtCompileTime>::run(
                dst, lhs.nestedExpression(), rhs, alpha);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag>
    struct generic_product_impl<Lhs, Rhs, DenseShape, SelfAdjointShape, ProductTag>
        : generic_product_impl_base<Lhs, Rhs, generic_product_impl<Lhs, Rhs, DenseShape, SelfAdjointShape, ProductTag>>
    {
        typedef typename Product<Lhs, Rhs>::Scalar Scalar;

        template <typename Dest> static void scaleAndAddTo(Dest& dst, const Lhs& lhs, const Rhs& rhs, const Scalar& alpha)
        {
            selfadjoint_product_impl<Lhs, 0, Lhs::IsVectorAtCompileTime, typename Rhs::MatrixType, Rhs::Mode, false>::run(
                dst, lhs, rhs.nestedExpression(), alpha);
        }
    };

    /***************************************************************************
* Diagonal products
***************************************************************************/

    template <typename MatrixType, typename DiagonalType, typename Derived, int ProductOrder> struct diagonal_product_evaluator_base : evaluator_base<Derived>
    {
        typedef typename ScalarBinaryOpTraits<typename MatrixType::Scalar, typename DiagonalType::Scalar>::ReturnType Scalar;

    public:
        enum
        {
            CoeffReadCost = int(NumTraits<Scalar>::MulCost) + int(evaluator<MatrixType>::CoeffReadCost) + int(evaluator<DiagonalType>::CoeffReadCost),

            MatrixFlags = evaluator<MatrixType>::Flags,
            DiagFlags = evaluator<DiagonalType>::Flags,

            _StorageOrder =
                (Derived::MaxRowsAtCompileTime == 1 && Derived::MaxColsAtCompileTime != 1) ?
                    RowMajor :
                    (Derived::MaxColsAtCompileTime == 1 && Derived::MaxRowsAtCompileTime != 1) ? ColMajor : MatrixFlags & RowMajorBit ? RowMajor : ColMajor,
            _SameStorageOrder = _StorageOrder == (MatrixFlags & RowMajorBit ? RowMajor : ColMajor),

            _ScalarAccessOnDiag =
                !((int(_StorageOrder) == ColMajor && int(ProductOrder) == OnTheLeft) || (int(_StorageOrder) == RowMajor && int(ProductOrder) == OnTheRight)),
            _SameTypes = is_same<typename MatrixType::Scalar, typename DiagonalType::Scalar>::value,
            // FIXME currently we need same types, but in the future the next rule should be the one
            //_Vectorizable = bool(int(MatrixFlags)&PacketAccessBit) && ((!_PacketOnDiag) || (_SameTypes && bool(int(DiagFlags)&PacketAccessBit))),
            _Vectorizable = bool(int(MatrixFlags) & PacketAccessBit) && _SameTypes &&
                            (_SameStorageOrder || (MatrixFlags & LinearAccessBit) == LinearAccessBit) &&
                            (_ScalarAccessOnDiag || (bool(int(DiagFlags) & PacketAccessBit))),
            _LinearAccessMask = (MatrixType::RowsAtCompileTime == 1 || MatrixType::ColsAtCompileTime == 1) ? LinearAccessBit : 0,
            Flags = ((HereditaryBits | _LinearAccessMask) & (unsigned int)(MatrixFlags)) | (_Vectorizable ? PacketAccessBit : 0),
            Alignment = evaluator<MatrixType>::Alignment,

            AsScalarProduct = (DiagonalType::SizeAtCompileTime == 1) ||
                              (DiagonalType::SizeAtCompileTime == Dynamic && MatrixType::RowsAtCompileTime == 1 && ProductOrder == OnTheLeft) ||
                              (DiagonalType::SizeAtCompileTime == Dynamic && MatrixType::ColsAtCompileTime == 1 && ProductOrder == OnTheRight)
        };

        EIGEN_DEVICE_FUNC diagonal_product_evaluator_base(const MatrixType& mat, const DiagonalType& diag) : m_diagImpl(diag), m_matImpl(mat)
        {
            EIGEN_INTERNAL_CHECK_COST_VALUE(NumTraits<Scalar>::MulCost);
            EIGEN_INTERNAL_CHECK_COST_VALUE(CoeffReadCost);
        }

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Scalar coeff(Index idx) const
        {
            if (AsScalarProduct)
                return m_diagImpl.coeff(0) * m_matImpl.coeff(idx);
            else
                return m_diagImpl.coeff(idx) * m_matImpl.coeff(idx);
        }

    protected:
        template <int LoadMode, typename PacketType> EIGEN_STRONG_INLINE PacketType packet_impl(Index row, Index col, Index id, internal::true_type) const
        {
            return internal::pmul(m_matImpl.template packet<LoadMode, PacketType>(row, col), internal::pset1<PacketType>(m_diagImpl.coeff(id)));
        }

        template <int LoadMode, typename PacketType> EIGEN_STRONG_INLINE PacketType packet_impl(Index row, Index col, Index id, internal::false_type) const
        {
            enum
            {
                InnerSize = (MatrixType::Flags & RowMajorBit) ? MatrixType::ColsAtCompileTime : MatrixType::RowsAtCompileTime,
                DiagonalPacketLoadMode =
                    EIGEN_PLAIN_ENUM_MIN(LoadMode, ((InnerSize % 16) == 0) ? int(Aligned16) : int(evaluator<DiagonalType>::Alignment))  // FIXME hardcoded 16!!
            };
            return internal::pmul(m_matImpl.template packet<LoadMode, PacketType>(row, col),
                                  m_diagImpl.template packet<DiagonalPacketLoadMode, PacketType>(id));
        }

        evaluator<DiagonalType> m_diagImpl;
        evaluator<MatrixType> m_matImpl;
    };

    // diagonal * dense
    template <typename Lhs, typename Rhs, int ProductKind, int ProductTag>
    struct product_evaluator<Product<Lhs, Rhs, ProductKind>, ProductTag, DiagonalShape, DenseShape>
        : diagonal_product_evaluator_base<Rhs, typename Lhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>, OnTheLeft>
    {
        typedef diagonal_product_evaluator_base<Rhs, typename Lhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>, OnTheLeft> Base;
        using Base::coeff;
        using Base::m_diagImpl;
        using Base::m_matImpl;
        typedef typename Base::Scalar Scalar;

        typedef Product<Lhs, Rhs, ProductKind> XprType;
        typedef typename XprType::PlainObject PlainObject;
        typedef typename Lhs::DiagonalVectorType DiagonalType;

        enum
        {
            StorageOrder = Base::_StorageOrder
        };

        EIGEN_DEVICE_FUNC explicit product_evaluator(const XprType& xpr) : Base(xpr.rhs(), xpr.lhs().diagonal()) {}

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Scalar coeff(Index row, Index col) const { return m_diagImpl.coeff(row) * m_matImpl.coeff(row, col); }

#ifndef EIGEN_GPUCC
        template <int LoadMode, typename PacketType> EIGEN_STRONG_INLINE PacketType packet(Index row, Index col) const
        {
            // FIXME: NVCC used to complain about the template keyword, but we have to check whether this is still the case.
            // See also similar calls below.
            return this->template packet_impl<LoadMode, PacketType>(
                row, col, row, typename internal::conditional<int(StorageOrder) == RowMajor, internal::true_type, internal::false_type>::type());
        }

        template <int LoadMode, typename PacketType> EIGEN_STRONG_INLINE PacketType packet(Index idx) const
        {
            return packet<LoadMode, PacketType>(int(StorageOrder) == ColMajor ? idx : 0, int(StorageOrder) == ColMajor ? 0 : idx);
        }
#endif
    };

    // dense * diagonal
    template <typename Lhs, typename Rhs, int ProductKind, int ProductTag>
    struct product_evaluator<Product<Lhs, Rhs, ProductKind>, ProductTag, DenseShape, DiagonalShape>
        : diagonal_product_evaluator_base<Lhs, typename Rhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>, OnTheRight>
    {
        typedef diagonal_product_evaluator_base<Lhs, typename Rhs::DiagonalVectorType, Product<Lhs, Rhs, LazyProduct>, OnTheRight> Base;
        using Base::coeff;
        using Base::m_diagImpl;
        using Base::m_matImpl;
        typedef typename Base::Scalar Scalar;

        typedef Product<Lhs, Rhs, ProductKind> XprType;
        typedef typename XprType::PlainObject PlainObject;

        enum
        {
            StorageOrder = Base::_StorageOrder
        };

        EIGEN_DEVICE_FUNC explicit product_evaluator(const XprType& xpr) : Base(xpr.lhs(), xpr.rhs().diagonal()) {}

        EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const Scalar coeff(Index row, Index col) const { return m_matImpl.coeff(row, col) * m_diagImpl.coeff(col); }

#ifndef EIGEN_GPUCC
        template <int LoadMode, typename PacketType> EIGEN_STRONG_INLINE PacketType packet(Index row, Index col) const
        {
            return this->template packet_impl<LoadMode, PacketType>(
                row, col, col, typename internal::conditional<int(StorageOrder) == ColMajor, internal::true_type, internal::false_type>::type());
        }

        template <int LoadMode, typename PacketType> EIGEN_STRONG_INLINE PacketType packet(Index idx) const
        {
            return packet<LoadMode, PacketType>(int(StorageOrder) == ColMajor ? idx : 0, int(StorageOrder) == ColMajor ? 0 : idx);
        }
#endif
    };

    /***************************************************************************
* Products with permutation matrices
***************************************************************************/

    /** \internal
  * \class permutation_matrix_product
  * Internal helper class implementing the product between a permutation matrix and a matrix.
  * This class is specialized for DenseShape below and for SparseShape in SparseCore/SparsePermutation.h
  */
    template <typename ExpressionType, int Side, bool Transposed, typename ExpressionShape> struct permutation_matrix_product;

    template <typename ExpressionType, int Side, bool Transposed> struct permutation_matrix_product<ExpressionType, Side, Transposed, DenseShape>
    {
        typedef typename nested_eval<ExpressionType, 1>::type MatrixType;
        typedef typename remove_all<MatrixType>::type MatrixTypeCleaned;

        template <typename Dest, typename PermutationType>
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Dest& dst, const PermutationType& perm, const ExpressionType& xpr)
        {
            MatrixType mat(xpr);
            const Index n = Side == OnTheLeft ? mat.rows() : mat.cols();
            // FIXME we need an is_same for expression that is not sensitive to constness. For instance
            // is_same_xpr<Block<const Matrix>, Block<Matrix> >::value should be true.
            //if(is_same<MatrixTypeCleaned,Dest>::value && extract_data(dst) == extract_data(mat))
            if (is_same_dense(dst, mat))
            {
                // apply the permutation inplace
                Matrix<bool, PermutationType::RowsAtCompileTime, 1, 0, PermutationType::MaxRowsAtCompileTime> mask(perm.size());
                mask.fill(false);
                Index r = 0;
                while (r < perm.size())
                {
                    // search for the next seed
                    while (r < perm.size() && mask[r]) r++;
                    if (r >= perm.size())
                        break;
                    // we got one, let's follow it until we are back to the seed
                    Index k0 = r++;
                    Index kPrev = k0;
                    mask.coeffRef(k0) = true;
                    for (Index k = perm.indices().coeff(k0); k != k0; k = perm.indices().coeff(k))
                    {
                        Block<Dest, Side == OnTheLeft ? 1 : Dest::RowsAtCompileTime, Side == OnTheRight ? 1 : Dest::ColsAtCompileTime>(dst, k).swap(
                            Block < Dest,
                            Side == OnTheLeft ? 1 : Dest::RowsAtCompileTime,
                            Side == OnTheRight ? 1 : Dest::ColsAtCompileTime > (dst, ((Side == OnTheLeft) ^ Transposed) ? k0 : kPrev));

                        mask.coeffRef(k) = true;
                        kPrev = k;
                    }
                }
            }
            else
            {
                for (Index i = 0; i < n; ++i)
                {
                    Block<Dest, Side == OnTheLeft ? 1 : Dest::RowsAtCompileTime, Side == OnTheRight ? 1 : Dest::ColsAtCompileTime>(
                        dst, ((Side == OnTheLeft) ^ Transposed) ? perm.indices().coeff(i) : i)

                        =

                            Block < const MatrixTypeCleaned,
                        Side == OnTheLeft ? 1 : MatrixTypeCleaned::RowsAtCompileTime,
                        Side == OnTheRight ? 1 :
                                             MatrixTypeCleaned::ColsAtCompileTime > (mat, ((Side == OnTheRight) ^ Transposed) ? perm.indices().coeff(i) : i);
                }
            }
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape> struct generic_product_impl<Lhs, Rhs, PermutationShape, MatrixShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs)
        {
            permutation_matrix_product<Rhs, OnTheLeft, false, MatrixShape>::run(dst, lhs, rhs);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape> struct generic_product_impl<Lhs, Rhs, MatrixShape, PermutationShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs)
        {
            permutation_matrix_product<Lhs, OnTheRight, false, MatrixShape>::run(dst, rhs, lhs);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
    struct generic_product_impl<Inverse<Lhs>, Rhs, PermutationShape, MatrixShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Inverse<Lhs>& lhs, const Rhs& rhs)
        {
            permutation_matrix_product<Rhs, OnTheLeft, true, MatrixShape>::run(dst, lhs.nestedExpression(), rhs);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
    struct generic_product_impl<Lhs, Inverse<Rhs>, MatrixShape, PermutationShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Inverse<Rhs>& rhs)
        {
            permutation_matrix_product<Lhs, OnTheRight, true, MatrixShape>::run(dst, rhs.nestedExpression(), lhs);
        }
    };

    /***************************************************************************
* Products with transpositions matrices
***************************************************************************/

    // FIXME could we unify Transpositions and Permutation into a single "shape"??

    /** \internal
  * \class transposition_matrix_product
  * Internal helper class implementing the product between a permutation matrix and a matrix.
  */
    template <typename ExpressionType, int Side, bool Transposed, typename ExpressionShape> struct transposition_matrix_product
    {
        typedef typename nested_eval<ExpressionType, 1>::type MatrixType;
        typedef typename remove_all<MatrixType>::type MatrixTypeCleaned;

        template <typename Dest, typename TranspositionType>
        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void run(Dest& dst, const TranspositionType& tr, const ExpressionType& xpr)
        {
            MatrixType mat(xpr);
            typedef typename TranspositionType::StorageIndex StorageIndex;
            const Index size = tr.size();
            StorageIndex j = 0;

            if (!is_same_dense(dst, mat))
                dst = mat;

            for (Index k = (Transposed ? size - 1 : 0); Transposed ? k >= 0 : k < size; Transposed ? --k : ++k)
                if (Index(j = tr.coeff(k)) != k)
                {
                    if (Side == OnTheLeft)
                        dst.row(k).swap(dst.row(j));
                    else if (Side == OnTheRight)
                        dst.col(k).swap(dst.col(j));
                }
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
    struct generic_product_impl<Lhs, Rhs, TranspositionsShape, MatrixShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs)
        {
            transposition_matrix_product<Rhs, OnTheLeft, false, MatrixShape>::run(dst, lhs, rhs);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
    struct generic_product_impl<Lhs, Rhs, MatrixShape, TranspositionsShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Rhs& rhs)
        {
            transposition_matrix_product<Lhs, OnTheRight, false, MatrixShape>::run(dst, rhs, lhs);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
    struct generic_product_impl<Transpose<Lhs>, Rhs, TranspositionsShape, MatrixShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Transpose<Lhs>& lhs, const Rhs& rhs)
        {
            transposition_matrix_product<Rhs, OnTheLeft, true, MatrixShape>::run(dst, lhs.nestedExpression(), rhs);
        }
    };

    template <typename Lhs, typename Rhs, int ProductTag, typename MatrixShape>
    struct generic_product_impl<Lhs, Transpose<Rhs>, MatrixShape, TranspositionsShape, ProductTag>
    {
        template <typename Dest> static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void evalTo(Dest& dst, const Lhs& lhs, const Transpose<Rhs>& rhs)
        {
            transposition_matrix_product<Lhs, OnTheRight, true, MatrixShape>::run(dst, rhs.nestedExpression(), lhs);
        }
    };

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_PRODUCT_EVALUATORS_H
